52 research outputs found
Modulation Equations: Stochastic Bifurcation in Large Domains
We consider the stochastic Swift-Hohenberg equation on a large domain near
its change of stability. We show that, under the appropriate scaling, its
solutions can be approximated by a periodic wave, which is modulated by the
solutions to a stochastic Ginzburg-Landau equation. We then proceed to show
that this approximation also extends to the invariant measures of these
equations
Multiscale Analysis for SPDEs with Quadratic Nonlinearities
In this article we derive rigorously amplitude equations for stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise acts only on
the stable modes and for an appropriate scaling between the distance from
bifurcation and the strength of the noise. We show that, due to the presence of
two distinct timescales in our system, the noise (which acts only on the fast
modes) gets transmitted to the slow modes and, as a result, the amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers equation
forced by additive noise in the orthogonal subspace to its dominant modes. The
theory developed in the present article thus allows to explain theoretically
some recent numerical observations from [Rob03]
Singular limits for stochastic equations
We study singular limits of stochastic evolution equations in the interplay
of disappearing strength of the noise and increasing roughness of the noise, so
that the noise in the limit would be too rough to define a solution to the
limiting equations. Simultaneously, the limit is singular in the sense that the
leading order differential operator may vanish. Although the noise is
disappearing in the limit, additional deterministic terms appear due to
renormalization effects.
We give an abstract framework for the main error estimates, that first reduce
to bounds on a residual and in a second step to bounds on the stochastic
convolution. Moreover, we apply it to a singularly regularized Allen-Cahn
equation and the Cahn-Hilliard/Allen-Cahn homotopy.Comment: 23 pages, 36 reference
Motion of a droplet for the Stochastic mass conserving Allen-Cahn equation
We study the stochastic mass-conserving Allen-Cahn equation posed on a smoothly bounded domain of R2 with additive, spatially smooth, space-time noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the L2- and H1-norms, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales
Ordered 2D colloidal photonic crystals on gold substrates by surfactant-assisted fast-rate Dip coating
Surfactant induced ordering of 2D and 3D colloidal crystal photonic crystals is possible on metallic substrates by dip‐coating at fast rates (≈1 mm/min). Ordered monolayer opals on conductive gold‐coated silicon substrates behave as a 2D diffraction grating. The method allows high throughput, ordered colloidal crystal formation useful as nanomaterials templates for energy storage or functional materials
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